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To show:
(i) ΔAPD ≅ ΔCQB
(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram
Proof:
(i) In ΔAPD and ΔCQB,
DP = BQ (Given)
∠ADP = ∠CBQ (Alternate interior angles)
AD = BC (Opposite sides of a ||gm
Thus, ΔAPD ≅ ΔCQB (by SAS congruence rule)
(ii) since, ΔAPD ≅ ΔCQB.
AP = CQ ( by CPCT )
(iii) In ΔAQB and ΔCPD,
BQ = DP (Given)
∠ABQ = ∠CDP (Alternate interior angles)
AB = CD (Opposite sides of a ||gm)
Thus, ΔAQB ≅ ΔCPD (by SAS congruence rule)
(iv) AQ = CP (by CPCT as ΔAQB ≅ ΔCPD.)
(v) From (ii) and (iv),
AP=CQ & AQ=CP
it is clear that APCQ has equal opposite sides also it has equal opposite angles.
Hence,APCQ is a ||gm.
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✅ Verified Answer ✅
Parallelogram :
A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram
.A quadrilateral is a parallelogram if
i)Its opposite sides are equal
ii) its opposite angles are equal
iii) diagonals bisect each other
iv) a pair of opposite sides is equal and parallel.
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Given: ABCD is a parallelogram and P and Q are points on BD such that
DP=BQ
To show:
(i) ΔAPD ≅ ΔCQB
(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram
Proof:
(i) In ΔAPD and ΔCQB,
DP = BQ (Given)
∠ADP = ∠CBQ (Alternate interior angles)
AD = BC (Opposite sides of a ||gm
Thus, ΔAPD ≅ ΔCQB (by SAS congruence rule)
(ii) since, ΔAPD ≅ ΔCQB.
AP = CQ ( by CPCT )
(iii) In ΔAQB and ΔCPD,
BQ = DP (Given)
∠ABQ = ∠CDP (Alternate interior angles)
AB = CD (Opposite sides of a ||gm)
Thus, ΔAQB ≅ ΔCPD (by SAS congruence rule)
(iv) AQ = CP (by CPCT as ΔAQB ≅ ΔCPD.)
(v) From (ii) and (iv),
AP=CQ & AQ=CP
it is clear that APCQ has equal opposite sides also it has equal opposite angles.
Hence,APCQ is a ||gm.
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